Some aspects of (r,k)-parking functions

TL;DR

This paper explores the properties of (r,k)-parking functions, providing combinatorial interpretations, symmetric function bases, and dualities, extending classical parking function theory with new algebraic and combinatorial insights.

Contribution

It introduces and analyzes (r,k)-parking functions, offering new combinatorial interpretations, symmetric function bases, and duality results, expanding the understanding of parking function generalizations.

Findings

Combinatorial interpretation of coefficients of a power series involving Frobenius characteristics.

Development of a symmetric function basis related to (r,1)-parking functions.

Establishment of a duality for (r,k)-parking functions when k<0.

Abstract

An \emph{$(r,k)$-parking function} of length $n$ may be defined as a sequence $(a_1,\dots,a_n)$ of positive integers whose increasing rearrangement $b_1\leq\cdots\leq b_n$ satisfies $b_i\leq k+(i-1)r$. The case $r=k=1$ corresponds to ordinary parking functions. We develop numerous properties of $(r,k)$-parking functions. In particular, if $F_n^{(r,k)}$ denotes the Frobenius characteristic of the action of the symmetric group $\mathfrak{S}_n$ on the set of all $(r,k)$-parking functions of length $n$, then we find a combinatorial interpretation of the coefficients of the power series $\left( \sum_{n\geq 0}F_n^{(r,1)}t^n\right)^k$ for any $k\in \mathbb{Z}$. When $k>0$, this power series is just $\sum_{n\geq 0} F_n^{(r,k)} t^n$; when $k<0$, we obtain a dual to $(r,k)$-parking functions. We also give a $q$-analogue of this result. For fixed $r$, we can use the symmetric functions $F_n^{(r,1)}$ to define a multiplicative basis for the ring $\Lambda$ of symmetric functions. We investigate some of the properties of this basis.